Proton momentum distribution

Let compare the preceding ECS and NCS results from formvar [Chatzidimitriou-Dreismann 2003 (a)]. A typical NCS TOF-spectrum is presented in Fig. 12. These results, obtained at T fa 295 K, demonstrate two novel features (i) the ECS technique provides proton momentum distributions which are in quantitative agreement with those obtained with NCS (see Fig. 13) ... 

To extract the proton mean kinetic energy, Ey), and the proton momentum distribution n(p),a general expression of the response function in Hermite polynomials H (x) is used. This expression can be written as... 

Measurements of the water proton mean kinetic energy and of the proton momentum distribution provide a richness of information about the potential surface that the proton experiences [3], including the effects of hydrogen bonding, thus complementing microscopic structural studies [7-9] and allowing a direct comparison with quantum Monte Carlo simulations [10,11]. 

The low temperature energy excess (Fig. 1), along with the bimodal proton momentum distribution (Fig. 2), is therefore a clear signature of quantum effects in supercooled water, and it is fair to state that the characteristic temperature dependence of the proton ( ] ) represents yet another example of the anomalous behavior of water [2], mostly evident in the supercooling region. 

Figure 4. Spherically averaged momentum distribution of water protons (4 r n(p)) at constant temperature T = 268K and several pressure values in the range 0.1-400 MPa. Note the absence of the high-momentum peak for supercooled water under pressure. The inset shows the high-momentum region of the distribution to evidence the absence of correlation between the applied pressure and the intensity of the tail of the proton momentum distribution.

Figure 8. Longitudinal momentum distribution for single ionization of helium by 945-keV antiproton (data points) in comparison with proton collision (full curve), (a) Electron momentum data [26] (b) recoil-ion data [26], The theoretical calculations represent antiproton collisions dotted curve, CDW results [26] broken curve, CTMC result [26], Here pze and pzr are equivalent to the notation of pey and pRy of Figs. 1 and 2, respectively.

The momentum distribution of H is derived from the measured NCS TOF-spectra by standard procedures [Mayers 1994 Mayers 2004], The distribution J(y) (often called "Compton profile" [Sears 1984 Watson 1996]) is proportional to the density of protons with momentum component hy along the direction of the neutron-proton momentum transfer hq. J(y) at scattering angle 0 = 66° is shown in Fig. 13 (full line). Here hy is the H-momentum component (before collision) along the direction of momentum transfer hq. 

Momentum distributions of protons are derived from ECS spectra. Fig. 13 shows the four measured distributions J(y) obtained from ECS with electrons of 15, 20, 25 and 30 keV, together with NCS data. These energies correspond to momentum transfers of 47.6, 55.1, 61.8 and 67.8 A-1, respectively. The agreement, within experimental error, confirms that both experiments reveal the same physical quantity the (projection on the scattering vector q of the) momentum density distribution of protons. The NCS and ECS results of Fig. 13 can be compared directly as the experimental energy resolution contributes negligibly to the width of the spectra obtained by other technique. 

Fig. 9. The momentum distribution of ejected electrons in collisions of 3 MeV protons with He atom for different angles of deflection of the projectile...

The momentum distributions thus obtained were applied to calculate inner-shell ionization cross sections by charged-particle impact in the BEA. For K-shell ionization process of Au by protons, the relativistic effect enhances the ionization cross sections significantly, and the relativistic BEA cross sections agree with the experimental values. On the other hand, the wave-function effect is negligible. 

For p d l the quantity exp(ip d) would be equal to unity and one would recover the standard cross section. The average < exp(ip-d) > depends on the initial momentum p of the proton and its orientation relative to H-H vector d. For p perpendicular to d, exp(ip-d)j = 1, but for p 11 d, the < exp(ip-d) >n terms are strongly reduced if there is a large zero-point contribution to the momentum distribution n(p). The oscillations in exp(ip-d) are effectively averaged out by the large intrinsic zero-point momentum spread in of the hydrogen isotopes, which typically amounts to A = 4 for protons (see Fig. 22.6). 

Fig. 14. The energy spectrum of protons quasi-elastically scattered from Be. The experimental arrangement is the same as that described in Fig. 13. The curves represent the spectrum calculated Cxussian momentum distributions with a 1/e value of 25 Mev for the solid line, and a 1/e value of 16 Mev for the dashed line.

Winsberg s yield of 2.8 day Sb requires the nucleus to have been left with an excitation energy well above 50 Mev a small percentage of the time. This energy is above the excitation energy which could be acquired if the protons in P27 really had a Fermi momentum distribution. The same conclusion can be drawn from the observations of Niklas and Lauterjung they found that some of the neutrons had energies of 45 Mev. 

Figure 2. Spherically averaged momentum distribution (in nip)) of water protons at several temperatures. Black solid and dashed lines refer to measurements in the supercooled metastable phase, while dash-dotted line is the result of a measurement at 298K. Note the appearance, in the supercooled phase data, of a peak or shoulder at high p, around p = n A , indicating proton coherent delocalization over two sites of the potential felt by protons. Experimental uncer tainties are less than 1%.

Figure 3. Top panel spherically averaged momentum distribution (4 r n(p)) of deuterons at T = 292.15K (solid line) and T = 276.15 K (dashed line). Bottom panel spherically averaged momentum distribution 4jr n p)) of deuterons at T = 276.15K (dashed line) compared to that of protons at T = 269.15K (black line), according to the shift of 7K due to the temperature difference between the density maxima of the two liquids.

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